Ads
related to: 2nd order elliptic differential equations
Search results
Results from the WOW.Com Content Network
The Poisson equation is a slightly more general second-order linear elliptic PDE, in which f is not required to vanish. For both of these equations, the ellipticity constant θ can be taken to be 1. The terminology elliptic partial differential equation is not used consistently throughout the literature
The Mathieu equation is a linear second-order differential equation with periodic coefficients. For q = 0, it reduces to the well-known harmonic oscillator, a being the square of the frequency. [4] The solution of the Mathieu equation is the elliptic-cylinder harmonic, known as Mathieu functions. They have long been applied on a broad scope of ...
In the theory of partial differential equations, elliptic operators are differential operators that generalize the Laplace operator. They are defined by the condition that the coefficients of the highest-order derivatives be positive, which implies the key property that the principal symbol is invertible, or equivalently that there are no real ...
Generalizing the maximum principle for harmonic functions which was already known to Gauss in 1839, Eberhard Hopf proved in 1927 that if a function satisfies a second order partial differential inequality of a certain kind in a domain of R n and attains a maximum in the domain then the function is constant. The simple idea behind Hopf's proof ...
Caffarelli, Nirenberg & Spruck (1985) has been particularly influential in the field of geometric analysis since many geometric partial differential equations are amenable to its methods. With Basilis Gidas, Spruck studied positive solutions of subcritical second-order elliptic partial differential equations of Yamabe type.
In mathematics, the p-Laplacian, or the p-Laplace operator, is a quasilinear elliptic partial differential operator of 2nd order. It is a nonlinear generalization of the Laplace operator , where p {\displaystyle p} is allowed to range over 1 < p < ∞ {\displaystyle 1<p<\infty } .
where is a second-order elliptic operator (implying that must be positive; a case where = + is considered below). A system of partial differential equations for a vector can also be parabolic. For example, such a system is hidden in an equation of the form
In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives.. The function is often thought of as an "unknown" that solves the equation, similar to how x is thought of as an unknown number solving, e.g., an algebraic equation like x 2 − 3x + 2 = 0.
Ads
related to: 2nd order elliptic differential equations